Expensive but falling as scale increases and government subsidies help drive scale.
Reliant on the grid - natural gas relies on the grid to move the air too. Not sure if this is the case all the time but mine won’t run at all without power, so unless you are referring to wood fired stove, natural gas furnaces are also reliant on the grid from what I can tell.
Gas/oil furnace can be powered by a small generator, solar panels or even a backup battery system, which is not the case for a heat pump. Oil / propane with local power generation can be fully off the grid.
Edit: the lower capital cost means you can also _afford_ a backup power supply.
Also, the largest heat pump in my house (I have a 6 kW and a 3 kW split system and a 5 kW air to water) also only uses about as much power as an electric kettle (~2 kW) most of the time here, I’m sure that could be powered off a small generator…
The blower motors are pretty power hungry, on the order of about 330 watts, give or take. For a 12V system that is about 30 amps, or about 2 hours of output from a typical deep cycle batter before you risk depleting it to the point of damage. An ~$300 12V Lifepo4 battery might get you 4 hours of runtime, but not much more than that. In most cases, an outage of less than 8 hours won't get things so cold in a decently insulated house that you need to worry about it, and IME, an outage that lasts more than 8 hours has high probability of lasting much much longer.
TL;DR - Your deep cycle battery approach may not prove to actually be worthwhile, consider working out the car-as-generator hookup plan in advance.
Ha yep, learned that the first time and was ready to go with an extension cord the second time I needed this setup. First time, I had a stash of 3 batteries but I didn't realize how much power the blower motor would draw, so I was charging on the car and swapping out batteries as needed. Did that the first day then rigged up cords for day 2. Halfway through day 3 we got power back.
Radiant in-floor heat is pretty efficient (and it's nice not to have an ice-cold floor first thing in the morning), just need power for the circulation pumps and the boiler. My back up heat is a wood-burning stove - and we have lost power for several hours when the temps are -20F or so...
If the boiler is in the basement would you not be able to treat the circulation tubes as a thermosiphon, with the hot water rising from the boiler, creating pressure to push water through the loop?
This is 2-4 hours of run time though. My experience is that the blower turns on for a period and then off for a period. It would depend on your heat loss rate which is situation specific, but you can probably get significantly more than this in time coverage with the blower only running periodically.
Natural gas heating relies on nothing more than convection to move hot water or (in older setups) steam from the boiler to the radiators. The only part of the setup where electricity is required is for electronic thermostats, and those can have a battery backup.